Sidon basis in polynomial rings over finite fields

Let ${\mathbb{F}}_q[t]$ denote the ring of polynomials over ${\mathbb{F}}_q$, the finite field of $q$ elements. Suppose the characteristic of ${\mathbb{F}}_q$ is not $2$ or $3$. In this paper, we prove an ${\mathbb{F}}_q[t]$-analogue of results related to the conjecture of Erd\H{o}s on the existence of infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We prove that there exists a $B_2[2]$ sequence of non-zero polynomials in ${\mathbb{F}}_q[t]$, which is an asymptotic basis of order $3$. We also prove that for any $\epsilon >0$, there exists a sequence of non-zero polynomials in ${\mathbb{F}}_q[t]$, which is a Sidon basis of order $3+\epsilon$. In other words, there exists a sequence of non-zero polynomials in ${\mathbb{F}}_q[t]$ such that any $n\in {\mathbb{F}}_q[t]$ of sufficiently large degree can be expressed as a sum of four elements of the sequence, where one of them has a degree less than or equal to $\epsilon \text{deg }n\mathrm{.}$